We provide a refinement of the Poincaré inequality on the torus \(\mathbb{T}^{d}\): there exists a set \(\mathcal{B} \subset \mathbb{T} ^{d}\) of directions such that for every \(\alpha \in \mathcal{B}\) there is a \(c_{\alpha } > 0\) with $$\begin{aligned} \|\nabla f\|_{L^{2}(\mathbb{T}^{d})}^{d-1} \| \langle \nabla f, \alpha \rangle \|_{L^{2}(\mathbb{T}^{d})} \geq c_{\alpha }\|f\| _{L^{2}(\mathbb{T}^{d})}^{d} \quad \mbox{for all}~f\in H^{1}\bigl( \mathbb{T}^{d}\bigr)~ \mbox{with mean 0.} \end{aligned}$$ The derivative \(\langle \nabla f, \alpha \rangle \) does not detect any oscillation in directions orthogonal to \(\alpha \), however, for certain \(\alpha \) the geodesic flow in direction \(\alpha \) is sufficiently mixing to compensate for that defect. On the two-dimensional torus \(\mathbb{T}^{2}\) the inequality holds for \(\alpha = (1, \sqrt{2})\) but is not true for \(\alpha = (1,e)\). Similar results should hold at a great level of generality on very general domains.

中文翻译：

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沿混合流的定向庞加莱不等式

我们对圆环\（\ mathbb {T} ^ {d} \）上的Poincaré不等式进行了细化：存在一组\（\ mathcal {B} \ subset \ mathbb {T} ^ {d} \）这样的方向，即每个\（\ alpha \ in \ mathcal {B} \）中都有一个\（c _ {\ alpha}> 0 \）与$$ \ begin {aligned} \ | \ nabla f \ | _ {L ^ {2}（\ mathbb {T} ^ {d}）} ^ {d-1} \ | \ langle \ nabla f，\ alpha \ rangle \ | _ {L ^ {2}（\ mathbb {T} ^ {d}）} \ geq c _ {\ alpha} \ | f \ | _ {L ^ {2}（\ mathbb {T} ^ {d}）} ^ {d} \ quad \ mbox {for all}〜f \ in H ^ {1} \ bigl（\ mathbb {T} ^ { d} \ bigr）〜\ mbox {均值为0。} \ end {aligned} $$导数\（\ langle \ nabla f，\ alpha \ rangle \）在正交于\（\ alpha \），但是，对于某些\（\ alpha \），方向\（\ alpha \）上的测地流已充分混合以补偿该缺陷。在二维圆环\（\ mathbb {T} ^ {2} \）上，不等式适用于\（\ alpha =（1，\ sqrt {2}）\），但不适用于\（\ alpha =（1，e）\）。在非常一般的领域中，类似的结果应具有很高的普遍性。